Properties of Monotonic Effects on Directed Acyclic Graphs

Various relationships are shown hold between monotonic effects and weak monotonic effects and the monotonicity of certain conditional expectations. Counterexamples are provided to show that the results do not hold under less restrictive conditions. Monotonic effects are furthermore used to relate signed edges on a causal directed acyclic graph to qualitative effect modification. The theory is applied to an example concerning the direct effect of smoking on cardiovascular disease controlling for hypercholesterolemia. Monotonicity assumptions are used to construct a test for whether there is a variable that confounds the relationship between the mediator, hypercholesterolemia, and the outcome, cardiovascular disease.

[1]  J. Pearl.  Causality: Models, Reasoning and Inference , 2000 .

[2]  D. A. Kenny,et al.  Correlation and Causation , 1937, Wilmott.

[3]  T. Haavelmo.  The Statistical Implications of a System of Simultaneous Equations , 1943 .

[4]  Dan Geiger,et al.  Identifying independence in bayesian networks , 1990, Networks.

[5]  Marek J. Druzdzel,et al.  Efficient Reasoning in Qualitative Probabilistic Networks , 1993, AAAI.

[6]  J. Pearl.  Causal diagrams for empirical research , 1995 .

[7]  P. Games.  Correlation and Causation: A Logical Snafu , 1990 .

[8]  Tyler J. VanderWeele,et al.  Four types of e ¤ ect modi cation-a classi cation based on directed acyclic graphs , 2007 .

[9]  Steffen L. Lauritzen,et al.  Independence properties of directed markov fields , 1990, Networks.

[10]  Linda C. van der Gaag,et al.  Monotonicity in Bayesian Networks , 2004, UAI.

[11]  P. Spirtes,et al.  Causation, prediction, and search , 1993 .

[12]  J. Robins,et al.  Signed directed acyclic graphs for causal inference , 2010, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[13]  Jin Tian,et al.  Polynomial Constraints in Causal Bayesian Networks , 2007, UAI.

[14]  Judea Pearl,et al.  Causal networks: semantics and expressiveness , 2013, UAI.

[15]  Judea Pearl,et al.  Direct and Indirect Effects , 2001, UAI.

[16]  A. P. Dawid,et al.  Corrigenda: Influence Diagrams for Causal Modelling and Inference , 2002 .

[17]  Michael P. Wellman.  Fundamental Concepts of Qualitative Probabilistic Networks , 1990, Artif. Intell..

[18]  J. Robins,et al.  Identifiability and Exchangeability for Direct and Indirect Effects , 1992, Epidemiology.

[19]  Jin Tian,et al.  On the Testable Implications of Causal Models with Hidden Variables , 2002, UAI.

[20]  James M. Robins,et al.  Empirical and counterfactual conditions for sufficient cause interactions , 2008 .

[21]  Jin Tian,et al.  Inequality Constraints in Causal Models with Hidden Variables , 2006, UAI.

[22]  J. Robins,et al.  Four Types of Effect Modification: A Classification Based on Directed Acyclic Graphs , 2007, Epidemiology.